Digital Signal Processing Reference
In-Depth Information
The evolution of the source queues is given in (11.1), while that of the cooperative queue is
+
coop
coop
CQ m Qm Dm
[
+=
1
]
(
[
]
−
[
])
+
VmHm
[ (
1
−
[
])
(
1
−
Vm CQ m
i
[
])
χ
(
[
])
.
(11.8)
i
i
j
j
j
{}
0
i
he queue states {
CQ
1
[
m
],
CQ
2
[
m
],
Q
1
[
m
],
Q
2
[
m
]}
m
=0
form a discrete-time Markov chain.
11.3.4
Stability Region of the Two-User Cooperative Slotted
ALOHA System
Due to the complex interactions among the four queues
Q
1
[
m
],
Q
2
[
m
],
CQ
1
[
m
], and
CQ
2
[
m
], the stability region of the cooperative system is difficult to obtain. However,
we are able to derive an inner bound by analyzing a dominant system where we let both
users be fully loaded, i.e.,
they always have a packet available for transmission. The sta-
bility region of this dominant system will be referred to as the
fully loaded region
in the
following.
When the users are fully loaded, the packet in
CB
i
will always be transmitted with
probability
-
i
as long as it is nonempty, i.e.,
CQ
i
[
m
] = 1. This is to say that when the users
are fully loaded, the states of the cooperative buffers are independent of the states of
the source buffers. In this case, {
CQ
1
[
m
],
CQ
2
[
m
]}
m
=0
forms a finite-state Markov chain
with four states:
S
0
= (0,0),
S
1
= (0,1),
S
2
= (1,0), and
S
3
= (1,1). For a given set of transmis-
sion probabilities
p
1
,
p
2
,
q
1
, and
q
2
, the Markov chain yields a steady-state distribution
denoted by the probabilities π
0
, π
1
, π
2
, and π
3
, where π
i
is the steady-state probability of
state
S
i
. The service rates for users 1 and 2, respectively, are given by
µ
(
ppqq
, , , =+
)
(
π π
)
qp
(
1
−
p
)
ψ
+− +π
[(
1
π
)] (
qp
1
−
p
)
ψ
,
11212
1
322
1
2
2
31 1
2
1
µ
(
ppqq
, , , =+
ppp
)
(
π π
)
q
(
1
−
)
ψ
+ −+
[(
1
π π
)] (
qp
1
−
p
)
ψ
.
21212
2
31
1
2
1
1
32 2
1
2
The stability region of the fully loaded system is
∪
R
=
R
(
p
,, ,, ,
pqq
212
)
(11.9)
coop
coop
1
2
(
pp
qq
, ∈,
, ∈,
)
[]
01
01
12
2
(
)
[]
12
where
R
coop
(
p
1
,
p
2
,
q
1
,
q
2
) = {(λ
1
,λ
2
) : λ
1
< μ
1
(
p
1
,
p
2
,
q
1
,
q
2
), λ
2
< μ
2
(
p
1
,
p
2
,
q
1
,
q
2
)} is the fully
loaded region given fixed values of
p
1
,
p
2
,
q
1
, and
q
2
. The region in (11.9) serves as an inner
bound to the stability region of the cooperative slotted ALOHA system.
11.3.5 Cooperative Slotted ALOHA with Channel Awareness
The cooperation considered up to this point exploits the diversity gains to improve the
throughput of the system. Since the same packet may be transmitted by both users in
the network, the probability of success will not be limited by the local channel quality
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